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   "source": [
    "Chapter 02\n",
    "\n",
    "# 向量夹角\n",
    "Book_4《矩阵力量》 | 鸢尾花书：从加减乘除到机器学习 (第二版)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8bb390c4-26af-44db-bf6a-d0edb2d14b1e",
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   "source": [
    "该代码定义了两个二维列向量 $a$ 和 $b$，并计算了它们之间夹角的余弦值、弧度和度数。\n",
    "\n",
    "### 计算公式\n",
    "\n",
    "向量 $a$ 和 $b$ 分别为：\n",
    "\n",
    "$$\n",
    "a = \\begin{bmatrix} 4 \\\\ 3 \\end{bmatrix}, \\quad b = \\begin{bmatrix} 5 \\\\ -2 \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "余弦夹角公式为：\n",
    "\n",
    "$$\n",
    "\\cos \\theta = \\frac{a \\cdot b}{\\|a\\|_2 \\|b\\|_2} = \\frac{a^T b}{\\|a\\|_2 \\|b\\|_2}\n",
    "$$\n",
    "\n",
    "接着，使用反余弦函数计算角度的弧度值：\n",
    "\n",
    "$$\n",
    "\\theta = \\arccos(\\cos \\theta)\n",
    "$$\n",
    "\n",
    "最后将弧度值转换为角度：\n",
    "\n",
    "$$\n",
    "\\theta_{\\text{degree}} = \\theta \\times \\frac{180}{\\pi}\n",
    "$$\n",
    "\n",
    "代码中的运算展示了向量间夹角的计算方法，包括余弦、弧度和度数转换。"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "01c7be45-8810-4f12-aca9-ecc099054269",
   "metadata": {},
   "source": [
    "## 导入所需库"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "53d8013b-3c6e-40da-bc98-e5787ed25f79",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np  # 导入NumPy库，用于数值计算"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6c657ddd-c8a1-4ddc-825f-e9e82bcade44",
   "metadata": {},
   "source": [
    "## 定义两个列向量"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "8e262906-50d0-44ee-b31b-4d6569b1b5b9",
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b = np.array([[4], [3]]), np.array([[5], [-2]])  # 定义向量a和b"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ff9561ac-02f3-4b67-82e7-be3dddfb4460",
   "metadata": {},
   "source": [
    "## 计算余弦值"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "50fba935-3aa3-41f7-8aed-f29544822233",
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "array([[0.51994695]])"
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     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "cos_theta = (a.T @ b) / (np.linalg.norm(a, 2) * np.linalg.norm(b, 2))  \n",
    "# 计算向量a和b之间夹角的余弦值\n",
    "cos_theta"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d65329e6-0d9e-4005-b6b0-a8c46fdb2c04",
   "metadata": {},
   "source": [
    "## 计算夹角的弧度值"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "a94cfe3e-0338-4fb4-99d2-0b2e5e08d198",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1.02400749]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cos_radian = np.arccos(cos_theta)  # 计算余弦值对应的弧度\n",
    "cos_radian"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "56bc6d0e-831a-4f2f-ab59-9ff9fd558a41",
   "metadata": {},
   "source": [
    "## 将弧度转换为度数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "f018e489-013e-4f38-bdb4-291023e46179",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[58.67130713]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cos_degree = cos_radian * (180 / np.pi)  # 将弧度转换为度数\n",
    "cos_degree"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "85a80909-2aac-49ed-bb7a-f8cc6b80ee7d",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ecd322f4-f919-4be2-adc3-69d28ef25e69",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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